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Exercise 111.35.8. Consider the morphism of schemes

\[ f : X = \mathop{\mathrm{Spec}}(\mathbf{F}_ p(t)) \longrightarrow \mathop{\mathrm{Spec}}(\mathbf{F}_ p(t^ p)) = S \]

Compute the tangent space of $X/S$ at the unique point of $X$. Isn't that weird? What do you think happens if you take the morphism of schemes corresponding to $\mathbf{F}_ p[t^ p] \to \mathbf{F}_ p[t]$?

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View Exercise 111.35.8 as pdf